Recovering Statistical Theory in the Context of Model Calibrations

Statistical theory has been relatively absent in the exercise of estimating parameters of an option pricing model from cross-sectional data at a fixed point of calendar time. The cross-sectional data typically consists of prices for options at various strikes and maturities at market close. The problem has been the formulation of an error model consistent with no arbitrage conditions satisfied by models and possibly also market data. The article presents such requisite error specifications consistent with no arbitrage conditions. The properties of such estimators are then analyzed on simulated data to evaluate biases in parameter estimates and their volatilities. The use of such error specifications coupled with maximum likelihood estimation makes available standard statistical tests for testing hypotheses on model parameter values. The methods proposed are finally illustrated on four popular models from the literature on data for options on the S&P 500 index at market close on April 30, 2013. It is shown that one can then conclude the risk neutral statistical significance of skewness, excess kurtosis, the presence of jump components, the negative correlation between volatility and the stock price, and the presence of volatility of volatility.